Cannot get rid of reported missing files
Blender won't remove the image because to it, the image is still in use. That is because the image has a 'fake' user, indicated by the 'F' next to the filename:



Add another background image (so you can access the image properties again, like I did above), choose your image and click the 'F' to remove the fake user. The image will now have 1 user, the background image, which you can then remove. After saving, closing and re-opening the file (or by clicking File> Save then File> Revert) the image will now be gone.

Fake Users are used for the very purpose that was causing you an issue, to keep data around even when it's not in use. This can be useful if you have multiple materials and you are swapping between them. The material might not currently in use, but you might want to keep it around in case you want to use it in future. You would click the 'F' to create a 'fake' user of the material data, making sure blender never deletes it, even if it's not in use.